THEORY OF COMPUTING, Volume 16 (11), 2020, pp. 1–8 www.theoryofcomputing.org NOTE On the Classical Hardness of Spoofing Linear Cross-Entropy Benchmarking Scott Aaronson∗ Sam Gunn Received November 1, 2019; Revised February 11, 2020; Published November 2, 2020 Abstract. Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: How hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen to prove a conditional hardness result for Linear XEB spoofing. Specifically, we show that the problem is classically hard, assuming that there is no efficient classical algorithm that, given a random n-qubit quantum circuit C, estimates the probability of C outputting a specific output string, say 0n, with mean squared error even slightly better than that of the trivial estimator that always estimates 1=2n. Our result automatically encompasses the case of noisy circuits. ∗Supported by a Vannevar Bush Fellowship from the US Department of Defense, a Simons Investigator Award, and the Simons “It from Qubit" collaboration. ACM Classification: F.1.3, F.1.2 AMS Classification: 81P68, 68Q17 Key words and phrases: quantum supremacy, quantum complexity, sampling complexity © 2020 Scott Aaronson, Sam Gunn cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/toc.2020.v016a011 SCOTT AARONSON,SAM GUNN 1 Introduction Quantum computational supremacy refers to the solution of a well-defined computational task by a programmable quantum computer in significantly less time than is required by the best known algorithms running on existing classical computers, for reasons of asymptotic scaling. It is a prerequisite for useful quantum computation, and is therefore seen as a major milestone in the field. The task of sampling from random quantum circuits (called RCS) is one proposal for achieving quantum supremacy [4, 5, 2]. Unlike other proposals such as Boson Sampling [1] and Commuting Hamiltonians [6], RCS involves a universal quantum computer – one theoretically capable of applying any unitary transformation. Furthermore, RCS currently appears to be the easiest proposal to implement at a large enough scale to demonstrate quantum supremacy. A research team based at Google has announced a demonstration of quantum computational supremacy, by sampling the output distributions of random quantum circuits [3]. To verify that their circuits were working correctly, they tested their samples using Linear Cross-Entropy Benchmarking (Linear XEB). This test simply checks that the observed samples tend to concentrate on the outputs that have higher probabilities under the ideal distribution for the given quantum circuit. More formally, given samples n n z1;:::;zk 2 f0;1g , Linear XEB entails checking that Ei[P(zi)] is greater than some threshold b=2 , where P(z) is the probability of observing z under the ideal distribution. In the regime of 40-50 qubits, these probabilities can be calculated by a classical supercomputer with enough time. While there is some support for the conjecture that no classical algorithm can efficiently sample from the output distribution of a random quantum circuit [5], less is known about the hardness of directly spoofing a test like Linear XEB. Results about the hardness of sampling are not quite results about the hardness of spoofing Linear XEB; a device could score well on Linear XEB while being far from correct in total variation distance by, for example, always outputting the items with the k highest probabilities. Under the assumption that the noise in the device is purely “depolarizing" – that a sample from the circuit was sampled correctly with probability b − 1 and otherwise sampled uniformly at random – there is stronger evidence that it is difficult to spoof Linear XEB. Namely, if there is a classical algorithm for sampling from a quantum circuit with perfectly depolarizing noise in time T, then with the help of an all-powerful but untrusted prover, one can calculate a good estimate for output probabilities in time 10T=(b − 1) with high probability over circuits. Together with results of [9], it follows that under the Strong Exponential Time Hypothesis there is a quantum circuit from which one cannot classically sample with depolarizing noise in time (b − 1)2(1−o(1))n [3]. We are unaware of any evidence that does not depend on such a strong assumption about the noise. However, Aaronson and Chen were able to prove the hardness of a different, related verification procedure from a strong hardness assumption they called the Quantum Threshold Assumption (QUATH) [2]. Informally, QUATH states that it is impossible for a polynomial-time classical algorithm to guess whether a specific output string like 0n has greater-than-median probability of being observed as the output of a given n-qubit quantum circuit, with success probability 1=2 + W(1=2n). They went on to investigate algorithms for breaking QUATH by estimating the output amplitudes of quantum circuits. For certain classes of circuits, output amplitudes can be efficiently calculated, but in general even efficiently sampling from the output distribution is impossible unless the polynomial hierarchy collapses [1, 6]. Aaronson and Chen found an algorithm for calculating amplitudes of arbitrary circuits that runs in time dO(n) and THEORY OF COMPUTING, Volume 16 (11), 2020, pp. 1–8 2 ON THE CLASSICAL HARDNESS OF SPOOFING LINEAR CROSS-ENTROPY BENCHMARKING poly(n;d) space, where d is the circuit depth. This is now used in some state-of-the-art simulations, but is still too slow and of the wrong form to violate QUATH for larger circuits, as there is no way to trade the accuracy for polynomial-time efficiency. Here, we formulate a slightly different assumption that we call XQUATH and show that it implies the hardness of spoofing Linear XEB. Like QUATH, the new assumption is quite strong, but makes no reference to sampling. In particular, while we don’t know a reduction, refuting XQUATH seems essentially as hard as refuting QUATH. Note that our result says nothing, one way or the other, about the possibility of improvements to algorithms for calculating amplitudes. It just says that there’s nothing particular to spoofing Linear XEB that makes it easier than nontrivially estimating amplitudes. Indeed, since the news of the Google group’s success broke, at least four results have potentially improved on the classical simulation efficiency, beyond what Google had considered. First, Gray and Kourtis were able to optimize tensor network contraction methods to obtain a faster classical amplitude estimator, though it is not competitive for calculating millions of amplitudes at once [7]. Second, Pednault et al. argued that, by using secondary storage, the largest existing classical supercomputers should be able to simulate the experiments done at Google in a few days [11]. Third, Napp et al. produced an efficient algorithm for approximately simulating average-case quantum circuits from a certain distribution of constant depth circuits, which is impossible to efficiently exactly simulate classically in the worst-case unless the polynomial hierarchy collapses [10]. This algorithm is not efficient for circuits as deep as those used by the Google team. Fourth, Zhou et al. used tensor network algorithms to simulate circuits as large as in the Google experiment, but with different 2-qubit gates that were easier to simulate [12]. Our result provides some explanation for why these improvements had to target the general problem of amplitude estimation, rather than doing anything specific to the problem of spoofing Linear XEB. 2 Preliminaries Throughout this note we will refer to random quantum circuits. Our results apply to circuits chosen from any distribution D over circuits on n qubits that is unaffected by appending NOT gates to any subset of the qubits at the end of the circuit.1 For every such distribution there is a corresponding version of XQUATH. For instance, we could consider a distribution where d alternating layers of random single- and neighboring two-qubit gates are applied to a square lattice of n qubits, similar to the Google experiment. Note that the actual distribution in the Google experiment might have been affected by appending NOT gates, but they could have applied random NOT gates to the end of their circuits classically and achieved the same fidelity. If circuits from D include randomly-chosen NOT gates in the final layer, then D obviously satisfies our condition. Our assumption XQUATH states that no efficient classical algorithm can estimate the probability of such a random circuit C outputting 0n, with mean squared error even slightly lower than the trivial algorithm that always estimates 1=2n. Definition 2.1 (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that takes as input a quantum circuit C D and produces an 1We will also assume that there is an efficient procedure for converting C D to a new, identically distributed, C0 with NOT gates applied to select outputs on C. THEORY OF COMPUTING, Volume 16 (11), 2020, pp. 1–8 3 SCOTT AARONSON,SAM GUNN n 2 estimate p of p0 = Pr[C outputs 0 ] such that 2 −n 2 −3n E[(p0 − p) ] = E[(p0 − 2 ) ] − W(2 ) where the expectations are taken over circuits C as well as the algorithm’s internal randomness. The simplest way to attempt to refute XQUATH might be to hope that C is near to a circuit that is classically simulable – e. g., if C contains only near-Clifford gates.